Prediction: one use of a fitted regression equation is to predict response-variable values for particular future...

  

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Prediction: one use of a fitted regression equation is to predict response-variable values for particular "future" combinations of explanatory-variable scores. Suppose, therefore, that we fit the model y = XB+ (with full assumptions on e), ob- taining the least squares estimate b of 3. Let x = [1,201,..., Tok] represent a set of explanatory-variable scores for which a prediction is desired, and let Yo be the (generally unknown, or not yet known) corresponding value of Y. The explanatory variable vector x does not necessarily correspond to an observation in the sample for which the model was fit. a) If we use Yo = xob to estimate E(Yo), then the error in estimation is d = Yo - E(Yo). Show that if the model is correct, then E(6) = 0 (i.e. Yo is an unbiased estimator of E(Yo)) and that V(6) = ²x (X'X)-¹x0. b) We may be interested not in estimating the expected value of Yo but in predicting or forecasting the actual value of Yo = xo3 +o that will be observed. The error in the forecast is then D = Yo-Yo = xob - (xo3 + o) = xo(b-3) - 50 Show that E(D) = 0 and that V(D) = o[1 + x(X'X)-¹xo]. Why is the forecast error D greater than the variance of 6 found in part (a)? c) Use the results in part (a) and (b), along with the Canadian occupational pres- tige regression (see Section 5.2.2), to predict the prestige score for an occupation with an average income of $12,000, an average education of 13 years, and 50% women. Place a 90% confidence interval around the predcition, assuming (i) that you with to estimate E(Yo) and (ii) that you wish to forecast an actual Yo score. (Because o is not known, you will need to use S2 and the f-distribution.) d) Suppose that the methods of this problem are used to forecast a value of Y for a combination of Xs very different from the X values in the data to which the model was fit. For example, calculate the estimated variance of the forecast error for an occupation with an average income of $50,000, an average education of 0 years, and 100% women. Is the estimated variance of the forecast error large or small? Does the variance of the forecast error adequately capture the uncertainty in using the regression equation to predict Y in this circumstance? Prediction: one use of a fitted regression equation is to predict response-variable values for particular "future" combinations of explanatory-variable scores. Suppose, therefore, that we fit the model y = XB+ (with full assumptions on e), ob- taining the least squares estimate b of 3. Let x = [1,201,..., Tok] represent a set of explanatory-variable scores for which a prediction is desired, and let Yo be the (generally unknown, or not yet known) corresponding value of Y. The explanatory variable vector x does not necessarily correspond to an observation in the sample for which the model was fit. a) If we use Yo = xob to estimate E(Yo), then the error in estimation is d = Yo - E(Yo). Show that if the model is correct, then E(6) = 0 (i.e. Yo is an unbiased estimator of E(Yo)) and that V(6) = ²x (X'X)-¹x0. b) We may be interested not in estimating the expected value of Yo but in predicting or forecasting the actual value of Yo = xo3 +o that will be observed. The error in the forecast is then D = Yo-Yo = xob - (xo3 + o) = xo(b-3) - 50 Show that E(D) = 0 and that V(D) = o[1 + x(X'X)-¹xo]. Why is the forecast error D greater than the variance of 6 found in part (a)? c) Use the results in part (a) and (b), along with the Canadian occupational pres- tige regression (see Section 5.2.2), to predict the prestige score for an occupation with an average income of $12,000, an average education of 13 years, and 50% women. Place a 90% confidence interval around the predcition, assuming (i) that you with to estimate E(Yo) and (ii) that you wish to forecast an actual Yo score. (Because o is not known, you will need to use S2 and the f-distribution.) d) Suppose that the methods of this problem are used to forecast a value of Y for a combination of Xs very different from the X values in the data to which the model was fit. For example, calculate the estimated variance of the forecast error for an occupation with an average income of $50,000, an average education of 0 years, and 100% women. Is the estimated variance of the forecast error large or small? Does the variance of the forecast error adequately capture the uncertainty in using the regression equation to predict Y in this circumstance?

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Here are the stepbystep workings for parts ad a If the model is correct then 1 Ed EYo EYo EYo EYo 0 ... View the full answer